Transcendental function evaluation

ABSTRACT

In described examples, an apparatus is arranged to generate a linear term, a quadratic term, and a constant term of a transcendental function with, respectively, a first circuit, a second circuit, and a third circuit in response to least significant bits of an input operand and in response to, respectively, a first, a second, and a third table value that is retrieved in response to, respectively, a first, a second, and a third index generated in response to most significant bits of the input operand. The third circuit is further arranged to generate a mantissa of an output operand in response to a sum of the linear term, the quadratic term, and the constant term.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of and claims priority to U.S. patent application Ser. No. 16/000,736, filed Jun. 5, 2018, which is incorporated by reference herein in its entirety.

BACKGROUND

A process for performing nonlinear control at high levels of performance can evaluate a transcendental function to generate a corrected error signal in a feedback control loop. The corrected error signal is produced in response to a control loop error. A function |x|^(α) is a transcendental function that generally uses substantial digital computation for its evaluation. In digital systems, such transcendental functions can be evaluated by determining a Taylor series expansion that consumes a high level of computing power and a substantial number of clock cycles for its execution. Hardware implementations for evaluation of transcendental functions have been proposed such as by using CORDIC (coordinate rotation digital computer), which is an iterative process that converges in accuracy during successive steps. Accordingly, the evaluation of a transcendental function such as |x|^(α) in a control loop can consume relative large amounts of power and time.

SUMMARY

In described examples, an apparatus is arranged to generate a linear term, a quadratic term, and a constant term of a transcendental function with, respectively, a first circuit, a second circuit, and a third circuit in response to least significant bits of an input operand and in response to, respectively, a first, a second, and a third table value that is retrieved in response to, respectively, a first, a second, and a third index generated in response to most significant bits of the input operand. The third circuit is further arranged to generate an output operand in response to a sum of the linear term, the quadratic term, and the constant term.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of an example system including an example execution unit for determining transcendental functions.

FIG. 2 is a block diagram of an example CPU including an example execution unit for determining transcendental functions.

FIG. 3 is a block diagram of an example FPU including an example execution unit for determining transcendental functions.

FIG. 4 is a block diagram of registers of example execution units for determining transcendental functions.

FIG. 5 is a high-level block diagram of an example execution unit for determining transcendental functions.

FIG. 6 is a block diagram of an example data path of an example logarithmic execution unit for determining logarithmic transcendental functions.

FIG. 7 is a block diagram of an example data path of an example exponentiation execution unit for determining exponential transcendental functions.

DETAILED DESCRIPTION

A system is described herein for evaluating a transcendental function such as |x|^(α). The transcendental function can be evaluated in response to curve-fitting the transcendental function over a sequence of equally or non-equally spaced segments, where the segments for such curve-fitting are determined relative to a mantissa of an input operand, and the curve-fitting generates an approximation of a value of the transcendental function. Examples of digital circuitry (such as an execution unit and/or hardware accelerator) are described herein for evaluating transcendental functions (such as a logarithmic or an exponential function) in response to linear or quadratic curve-fitting.

An example execution unit can be arranged to evaluate a transcendental function by quickly evaluating functions otherwise performed by a general purpose processor executing software (which otherwise executes over longer periods of time). The execution unit can operate in conjunction with a processing unit such as floating point unit (FPU) to provide improved speed, accuracy and suitability for real-time control applications. The example execution unit described herein can be emulated, for example, with programmable logic circuits and provide speed, accuracy, and suitability for real-time control applications.

For evaluating a logarithmic function of an input value, an input floating-point number is segregated into the constituent mantissa and exponent parts. Linear and quadratic terms of the transcendental function can be evaluated in response to curve-fitting the mantissa. The linear and quadratic terms of the curve-fitting can be shifted and combined with exponent values determined in response to the input exponent. The mantissa can be left-shifted for cases in which the exponent is zero and the mantissa has leading zeros. The final result can be left-shifted left to generate a floating-point number in proper form that includes an adjusted exponent.

For evaluating an inverse exponential function (which is an exponential function) of an input value, a input floating-point number is segmented in response to a slope of the function determined by the input mantissa and exponent. Linear and quadratic terms of the transcendental function can be evaluated in response to curve-fitting the mantissa over non-equally spaced segments. The linear and quadratic terms are combined to form a final result. Final results for input values having points near zero are determined in response to more higher resolution x-value components (e.g., as compared to the resolution of very large input values) in a curve fit due to the floating-point representation. Spacing the segments (and selecting the number of segments) in response to a slope of the function determined by the input exponent maintains the cardinality of samples for a selected accuracy over the range of input values when evaluating the exponential function.

Accordingly, the example execution unit for evaluating transcendental functions can be included in a digital implementation of a system, such as a real-time feedback system, which substantially reduces the latency of the time period for calculating the result of a transcendental function.

In an example system described hereinbelow, the function log₂|x| can be evaluated in six computer cycles to an accuracy of about 2⁻²³ when referencing a table of 128 segments, whereas an example FPU executing software can evaluate the function in as few as 35 computer cycles. The example system can evaluate in seven computer cycles the function 2^(−|x|) to an accuracy of about 2⁻²⁰ when referencing a table of 249 segments, whereas an example FPU executing software can evaluate in seven computer cycles the function in as few as 69 computer cycles. The example system can evaluate in eight computer cycles the function log_(e)|x|=log₂|x|*(1/(log₂(e)), whereas an example FPU executing software can evaluate the function in as few as 31 computer cycles. The example system can evaluate in nine computer cycles the function e^(x)=2^(x*log 2(e)), whereas an example FPU executing software can evaluate the function in as few as 43 computer cycles. The level of accuracy and the reduced latency provides sufficiently accurate and timely numerical results for purposes of a real-time nonlinear control application. In other systems designed according to the techniques of this disclosure, various combinations of more, less, or the same number of cycles may be used for evaluating one or more of the above-mentioned functions.

In the example described hereinbelow, an FPU is coupled to an example execution unit and is arranged to provide a floating point number and an indication of an instruction type to the execution unit. The FPU is arranged to wait for six computer cycles for log function results, and seven computer cycles for inverse exponentiation function results. However, the FPU is pipelined, such that the FPU need not remain idle while awaiting results. Accordingly, the latency of the FPU is reduced, such that the FPU can more quickly regulate a nonlinear feedback loop, for example.

In some examples, the execution unit is arranged as at least two pipeline stages in which the second stage determines a second-portion of an evaluation for a first input operand while the first stage determines a first-half of an evaluation for a second input operand. Accordingly, the throughput of the example pipelined hardware-accelerated system can be doubled as compared to an otherwise similar example non-pipelined hardware-accelerated system.

FIG. 1 is a block diagram of an example system 100 including an example hardware accelerator for determining transcendental functions. For example, the system 100 includes a to-be-controlled system (also referred to as a “controlled system”) 130. The system 100 includes a feedback path 160 for operating in response to output state signal 150 generated by the controlled system 130. The output state signal 150 is coupled to an inverting input of an adder 110 via the feedback path 160. The noninverting input of the adder 110 receives a target output state signal 140 that represents a target output state of the controlled system 130. In response to the output state signal 150 and the target output state signal, the adder 110 generates a state error signal 170 to be processed by a central processing unit (CPU) 120.

The CPU 120 includes a processor 122, memory 124, an FPU 126 and an execution unit 128 that generates a non-linearized state error signal 170 that is presented as a control input to the controlled system 130. The execution unit 128 is arranged to digitally compute the (e.g., non-linearized) state error signal 170. The state error signal 170 can be represented by a transcendental function |x|^(α), where x represents the state error signal 170 and α is a constant in a range that extends, for example, from 0.2 to 2.0. In such a manner, a non-linearized response of the controlled system 130 can be accurately generated by the execution unit 128. The processor 122 is arranged to generate an input operand for hardware accelerated calculation of a transcendental function, while the memory 124 is configured to receive and store the input operand. The execution unit 128 is arranged to generate as an approximation of a transcendental function in response to the input operand stored and retrieved from the memory 124.

FIG. 2 is a block diagram of an example CPU 200 including an example execution unit for determining transcendental functions. The CPU 200 can be a processor, such as the CPU 120. The CPU 200 includes an operand bus 220 coupled to receive data from a data read bus 210. The CPU 200 also includes a result bus 270 coupled to write data to a data write bus 280. Results of computation generated by the CPU 200 are asserted on the result bus 270.

The CPU 200 also includes an FPU 250 coupled (e.g., closely coupled) to an execution unit 260. The execution unit 260 (which can include a hardware accelerator and/or digital logic circuits as described further hereinbelow) is arranged to generate an approximation (e.g., close estimate) of a transcendental function result in response to an input operand stored and retrieved from register memory accessible by the FPU 250.

In an example multiplication operation not involving the execution unit, a “multiply” instruction is indicated to the FPU 250 by the CPU instruction controller 240. In response to the “multiply” instruction indication (and other control signals from the CPU instruction controller 240), the FPU 250 reads two floating-point numbers as input operands. The input operands can be stored in CPU registers 230 (and received by the FPU 250 via the data read bus 210) or received by the FPU 250 from external memory (via the data read bus 210 and the operand bus 220). In response to received operational codes (e.g., instructions), the CPU instruction controller 240 coordinates scheduling and execution of FPU-related instructions and operands between external memory, the CPU registers 230, and the FPU 250, for example.

FIG. 3 is a block diagram of an example FPU 300 including an example execution unit for determining transcendental functions. The FPU 300 can be a processor, such as the FPU 250. The FPU 300 includes an execution unit 370 arranged as a co-processor (e.g., with respect to a processor such as CPU 200). A register bank 310 (such as R1, R2, . . . , R7) is configured as a scratch-pad memory. The FPU 300 includes multiplier hardware 320 and adder hardware 330, which is arranged to execute floating-point arithmetic operations.

The FPU 300 includes the execution unit 370. The execution unit 370 is arranged to evaluate, at least, exponential and logarithmic transcendental functions. The execution unit 370 includes exponential hardware (EXP hardware) 340 that is arranged to evaluate exponential functions (including inverse exponential functions) and also includes logarithmic hardware (LOG hardware) 350 that is arrange to evaluate logarithmic functions.

A top-level controller 360 of the FPU 300 executes instructions in response to clocked operation of a state machine that is arranged to execute opcodes. For example, the top-level controller 360 generates and outputs bus control signals during a clock cycle for transferring information in response to a opcode received during a previous clock cycle. The top-level controller 360 can also operate in response to its own previous output. The FPU 300 is coupled to the execution unit 370, such that the FPU 300 and execution unit 370 are arranged to execute instructions (e.g., transcendental functions) more (e.g., much more) quickly than the FPU 300 could execute alone the same instruction (e.g., using firmware-encoded algorithms to sequence the operation of adders and multipliers).

FIG. 4 is a block diagram of registers and example registers of execution units for determining transcendental functions. The programming model 400 can include registers 420 of an FPU (which can be a processor such as the FPU 300) and registers 430 of execution units 440 (which can be an accelerator such as the execution unit 370). In the architecture described herein, the FPU is closely coupled to the execution units 440, which facilitates transfer of operands to and from the execution units 440.

The registers 420 (e.g., that form a portion of the FPU) can include latches and/or flip-flops for accessibly storing digital information. The registers 420 include registers R0, R1, . . . , R7 (respectively designated 421, 422, 423, and 424), a coefficient table 425, and flags 426 such as a flag LVF (overflow flag) and flag LUF (underflow flag). The flags 426 can indicate an underflow or an overflow condition encountered as a result of a computation executed by the FPU in response to a received operand. The registers 420 are used to store information for relatively quick internal (as compared to external memory, for example) access by the FPU.

A data bus 460 is a data read bus (from the perspective of the execution units 440) by which information (such as operands) stored in an FPU register (such as registers 420) can be read and stored in selected registers (e.g., at least one register 431, 432, 433, 434, 435, or 436) of the execution units 440. The data bus 450 is a data write bus (from the perspective of the execution units 440) by which selected registers of the execution unit registers 430 can be accessed and stored in at least one selected register of the FPU registers 420. The execution units 440 includes circuits (e.g., dedicated hardware) for evaluating a selected mathematic function that can be executed in hardware more quickly than a general purpose processor executing instructions for evaluating the respective mathematic function, for example.

The execution units 440 includes registers 430 arranged to store input and/or output operands for underlying execution units. The registers 430 are arranged to read and write (e.g., mathematic function input and/or output) operands, such as ADDF32 operands (32-bit floating point addition operands) 431, MPYF32 operands (32-bit floating point multiplication operands) 432, CMPF32 opcode (32-bit floating point comparator operands) 435, and ABSF32 operands (32-bit floating point absolute value operands) 436.

Additionally, the execution units 440 includes registers for accessibly storing operations for accelerating evaluations for transcendental operations. For example, 32-bit operands for floating point exponentiation (IEXP2F32) can be read into and/or written from IEXP2F32 register 433, and 32-bit operands for floating point base-2 logarithms (LOG 2F32) can be read into and/or written from LOG 2F32 register 434. The registers 433 and 444 are closely coupled to dedicated circuitry (such as a high-speed floating integer exponent engine and a floating logarithmic engine, respectively) of the execution units 440.

In an example IEXP2F32 operation, the execution units 440 can read an input operand via the data bus 460 from the registers 420, such that the input operand is stored in the register 433. The FPU can execute no-op opcodes, when not pipelined for example, to account for the time for the execution units 440 used by the execution units 440 to evaluate the exponential function. After the execution units 440 has evaluated the exponential of the input operand and has stored the output operand (e.g., result) in the register 433, the execution unit 440 writes the contents of the register 433 (via the data write bus 450) to the FPU, such that the FPU obtains the exponentiation results generated by the execution units 440.

In an example LOG 2F32 operation, the execution units 440 can read an input operand on via data bus 460 from the registers 420, such that the input operand is stored in the register 434. The FPU can execute no-op opcodes, when not pipelined for example, to account for the time for the execution units 440 used by the execution units 440 to evaluate the logarithmic function. After the execution units 440 has evaluated the logarithm of the input operand and has stored the output operand (e.g., result) in the register 434, the execution unit 440 writes the contents of the register 434 (via the data write bus 450) to the FPU, such that the FPU obtains the logarithmic results generated by the execution units 440.

In examples, the registers 432 and 433 can each store both the input and output operands (e.g., at a same time). Additionally, the registers 432 and 433 can each be arranged (e.g., duplicated) to store one or both of the input/output operands for overlapping, pipelined execution of two same-type (e.g., both exponentiation or both logarithmic) or different-type (e.g., one exponentiation and one logarithmic) transcendental functions.

FIG. 5 is a high-level block diagram of an example execution unit for determining transcendental functions. The circuit 500 is an accelerator, such as the execution units 440 described hereinabove. The circuit 500 is arranged to determine a value (e.g., estimated value) of a transcendental function, such as a logarithmic function or an exponential function. The type of transcendental function to be performed can be determined in response to the register in which the input operand is stored. The transcendental calculation is approximated with a quadratic curve-fitting operation, in which a quadratic equation of the form a*x²+b*x+c is evaluated over a segment of values in response to the input operand x.

The circuit 500 is operable as an execution unit configured to receive data from registers of an FPU and write data into the registers of the FPU. The circuit 500 and the FPU can be arranged to produce a non-linearized state error signal for controlling a system in response to (e.g., determining a difference between) an output state of the system and a target output state.

For example, the circuit 500 receives an input operand 510. The input operand 510 can be a floating point number, such that in the input operand 510 includes a mantissa 511, an exponent 514, and sign bits. The mantissa 511 includes (e.g., a set of) mantissa most significant bits (MSBs) 512 and (e.g., a set of) mantissa least significant bits (LSBs) 513. (The MSBs and the LSBs respectively are not necessarily the bits of the highest order or bits of the lowest order that are available: accordingly the term “most significant bits” can mean “more significant bits” and the term “least significant bits” can mean “less significant bits.”) The circuit 500 is configured to receive the input operand 510, which can be read from a register of the FPU. The circuit 500 is configured to generate a result in response to the input operand 510 and to write an output operand 590 into the registers of the FPU. The result is generated by evaluating (e.g., estimating) a quadratic equation in response to the input operand. In various examples, differing numbers of bits (and formats of real numbers) of input operands can be used in accordance with speed of calculation, complexity of circuitry, accuracy of output values, and combinations thereof.

A first circuit 530 is arranged to generate a linear term (e.g., “b*x”) of the transcendental function for curve-fitting. The linear term is generated in response to selected LSBs 513 of the mantissa 511 of the input operand 510 and in response to a first table value that is retrieved from a table(s) 520 in response to a first index generated in response to selected MSBs 512 of the mantissa 511. When the transcendental function being evaluated is an exponential function, the first index is also determined in response to the exponent 514.

A second circuit 550 is arranged to generate a quadratic term (e.g., “a*x²”) of the transcendental function for curve fitting. The quadratic term is generated in response to selected LSBs 513 of the mantissa 511 of the input operand 510 and in response to a second table value that is retrieved from a table(s) 520 in response to a second index generated in response to the MSBs 512 of the mantissa 511 of the input operand 510. When the transcendental function being evaluated is an exponential function, the second index is also determined in response to the exponent 514.

A third circuit 580 is arranged to generate (e.g., to output) a constant term (e.g., “c”) and to combine the linear and quadratic terms generated by the first circuit 530 and the second circuit 550. The constant term is generated for the transcendental function for the curve fit in response to the linear and quadratic terms and in response to a third table value that is retrieved from a table(s) 520 in response to a third index generated in response to the MSBs 512 of the mantissa 511. When the transcendental function being evaluated is an exponential function, the third index is also determined in response to the exponent 514. Additionally, the third circuit 580 is arranged to generate a mantissa of the output operand 590 in response to a sum of the linear term, the quadratic term, and the constant term. The table(s) 520 can be a unified table, or can be separated into a first table, a second table and a third table.

In examples, the first and second circuits 530 and 550 are arranged for parallel execution (e.g., where the linear term and the quadratic term are each determined during respective time periods that overlap in time). The first and second circuits 530 and 550 can be arranged as a first stage 501 in a pipeline, and the third circuit 580 can be arranged as a second stage 502 in the pipeline, such that the second stage 502 can add the quadratic, linear, and constant terms of a first operand (e.g., to be evaluated by a first transcendental function) during a first time interval that overlaps in time a second time interval in which the first stage 501 is determining the linear and quadratic terms in response to a second operand (e.g., to be evaluated by a second transcendental function). The first stage 501 and the second stage 502 that are arranged in a pipeline configuration facilitate overlapping execution of successive operations, such that throughput can be doubled (e.g., after the pipeline is filled).

The circuit 500 is programmable to selectively generate the output operand 590 as either a logarithmic result or an exponentiated result in response to a command generated by an external processor such as the FPU. The third circuit can select one of the logarithmic and exponential functions in response to a decoded instruction, for example. The logarithmic function can be evaluated in response to curve-fitting of table values indexed in response to the mantissa, whereas the exponentiation function can be evaluated in response to a segmented curve-fitting approach in which the domain of values of a table is divided into segments (e.g., non-equally spaced segments) and the table is indexed in response to the curve-fitting of each indexed segment.

For the exponentiation function, a table (such as Table 2 described hereinbelow) includes sequences of non-equally spaced segments of values for approximating results of a the function in response to an index derived in response to the mantissa 511 and exponent 514. Of the input operand 510. The first table value retrieved from the first table in response to the first index, the second table value retrieved from the second table in response to the second index, and the third table value retrieved from the third table in response to the third index are values associated with endpoints of each of the non-equally spaced segments of the table of values. As an example, evaluation of the transcendental function (|x|^(α)) for different x and a values can be generated by determining intermediate values Z1 and Z2 from which to the final result Z3=(|x|^(α)) can be determined:

Z1=log₂ |x|∀x∈[−∞,∞]  (1)

Z2=αZ1∀Z1∈(|∞,0)  (2)

Z3=2^(−|Z2|) ∀Z2∈(−∞,∞)  (3)

Accordingly, both logarithmic and exponential functions can evaluated in accordance with the intermediate values Z1, Z2, and Z3 determined for the transcendental function (|x|^(α)).

Evaluating transcendental functions (such as the exponential function 2^(x)) can yield different accuracies for a given amount of computation. For example, the slope of the exponential function 2^(x) increases exponentially in response to a given increase in the x value. The density of floating point number is highest where x is near zero and the density decreases on either side of the number line as x diverges from the zero point. These two nonlinear effects cause the exponential function output to have a maximum density (e.g., a lesser range of y-values for a given range of x-values) around the zero point, and lower density (e.g., a greater range of y-values for a give range of x-values) further from the zero point. The curve-fitting techniques described herein for evaluating exponentiation functions maintain accuracy over the domain of values in response to curve-fitting values from non-equally spaced table segments, for example. The (e.g., ordinate) spacing between values in the domain of table values for curve-fitting is determined based on (e.g., the slope of the function of) the exponent at a point (e.g., for a given exponent value). Optimal (e.g., for a target application) spacing for each interval can be determined in response to least-mean-square analysis of a regression of interval spacing and table lengths to determine results of sufficient accuracy (e.g., cardinality) and table length (e.g., the number of indexed entries). Table 1 includes the number of entries in the table for each exponent in the range −18 to 5.

TABLE 1 EXP ENTRIES −18 1 −17 1 −16 1 −15 1 −14 1 −13 1 −12 1 −11 1 −10 1 −9 1 −8 1 −7 2 −6 4 −5 8 −4 8 −3 16 −2 32 −1 32 0 32 1 32 2 32 3 32 4 8 5 0

Example tables are set forth in Table 2 hereinbelow for evaluating the IEXP2F32 and LOG 2F32 functions. The coefficients for the curve-fit approach can be determined based on a least-mean-square approach.

The notation “IEXP2F32” and “LOG 2F32” in the example tables below represent the macro names of extended instructions (opcodes) for computing base-2, 32-bit floating-point exponential or logarithmic results. The table entries are represented as hexadecimal numbers. An “SL” is the slice/index number for accessing a table entry; Y0i is a constant value for a quadratic fit for the transcendental function; the terms “S1i” and the “S2i” are, respectively, the linear and quadratic term coefficients. In the example, Table 2 includes 249 slices, each of which can be accessed using an index that varies from 1 to 249. For a logarithmic table, the spacing of the slices (e.g., along the x-axis of the functions of Table 2) is constant, and for an exponential table, the spacing of slices (e.g., along the x-axis of the functions of Table 2) is variable.

TABLE 2 SL NO: Y0i S1i S2i IEXP2F32 Tables 1 0xFFFFBD74 0x000B00 0x00000 2 0xFFFF7AEA 0x001640 0x00000 3 0xFFFEF5D4 0x002C40 0x00000 4 0xFFFDEBAB 0x0058C0 0x00000 5 0xFFFBD75B 0x00B180 0x00000 6 0xFFF7AEC8 0x0162C0 0x00000 7 0xFFEF5DD7 0x02C580 0x00040 8 0xFFDEBCC4 0x058AC0 0x00100 9 0xFFBD7DDB 0x0B1440 0x003C0 10 0xFF7B0CFF 0x1622C0 0x00F40 11 0xFEF65F0A 0x2C2E80 0x03D40 12 0xFE45E242 0x2C0FE0 0x03D20 13 0xFD95DFA5 0x2BF160 0x03D00 14 0xFCE656DE 0x2BD300 0x03CC0 15 0xFC374798 0x2BB4B0 0x03CA0 16 0xFB88B180 0x2B9660 0x03C70 17 0xFADA9442 0x2B7840 0x03C40 18 0xFA2CEF8A 0x2B5A28 0x03C18 19 0xF97FC304 0x2B3C20 0x03BF0 20 0xF8D30E5E 0x2B1E38 0x03BC8 21 0xF826D144 0x2B0060 0x03BA0 22 0xF77B0B64 0x2AE298 0x03B78 23 0xF6CFBC6B 0x2AC4E8 0x03B50 24 0xF624E407 0x2AA750 0x03B20 25 0xF57A81E5 0x2A89C8 0x03AF8 26 0xF47BCBBD 0x54BB48 0x0EAF0 27 0xF329C922 0x544628 0x0E9A8 28 0xF1D999D8 0x53D1A0 0x0E868 29 0xF08B3B58 0x535DC0 0x0E728 30 0xEF3EAB20 0x52EA80 0x0E5E8 31 0xEDF3E6B1 0x5277D8 0x0E4A8 32 0xECAAEB8F 0x5205D8 0x0E368 33 0xEB63B742 0x519470 0x0E230 34 0xEA1E4755 0x5123A8 0x0E0F8 35 0xE8DA9957 0x50B378 0x0DFC0 36 0xE798AADA 0x5043E8 0x0DE8C 37 0xE6587973 0x4FD4F0 0x0DD58 38 0xE51A02BA 0x4F6690 0x0DC24 39 0xE3DD444B 0x4EF8C8 0x0DAF4 40 0xE2A23BC7 0x4E8B9C 0x0D9C8 41 0xE168E6CF 0x4E1F04 0x0D898 42 0xE0314309 0x4DB300 0x0D76C 43 0xDEFB4E1F 0x4D4794 0x0D644 44 0xDDC705BC 0x4CDCBC 0x0D51C 45 0xDC946790 0x4C7278 0x0D3F4 46 0xDB63714F 0x4C08C8 0x0D2D0 47 0xDA3420AD 0x4B9FA8 0x0D1AC 48 0xD9067364 0x4B371C 0x0D08C 49 0xD7DA6730 0x4ACF20 0x0CF6C 50 0xD6AFF9D1 0x4A67B0 0x0CE4C 51 0xD5872909 0x4A00D2 0x0CD2E 52 0xD45FF29D 0x499A82 0x0CC12 53 0xD33A5457 0x4934C0 0x0CAF8 54 0xD2164C01 0x48CF8A 0x0C9E0 55 0xD0F3D76C 0x486AE0 0x0C8C8 56 0xCFD2F467 0x4806C0 0x0C7B4 57 0xCEB3A0CA 0x47A32C 0x0C6A0 58 0xCD95DA6A 0x474022 0x0C58C 59 0xCC799F23 0x46DDA0 0x0C47C 60 0xCB5EECD3 0x467BA6 0x0C36C 61 0xCA45C15A 0x461A34 0x0C25E 62 0xC92E1A9D 0x45B948 0x0C150 63 0xC817F680 0x4558E4 0x0C046 64 0xC70352EF 0x44F902 0x0BF3C 65 0xC5F02DD6 0x4499A8 0x0BE34 66 0xC4DE8523 0x443AD0 0x0BD2C 67 0xC3CE56C9 0x43DC7A 0x0BC26 68 0xC2BFA0BC 0x437EA8 0x0BB22 69 0xC1B260F5 0x432158 0x0BA20 70 0xC0A6956E 0x42C488 0x0B91E 71 0xBF9C3C24 0x426838 0x0B81E 72 0xBE935317 0x420C6A 0x0B720 73 0xBD8BD84B 0x41B118 0x0B624 74 0xBC85C9C5 0x415646 0x0B528 75 0xBB81258D 0x40FBF2 0x0B42C 76 0xBA7DE9AE 0x40A21A 0x0B334 77 0xB97C1437 0x4048BE 0x0B23C 78 0xB87BA337 0x3FEFDE 0x0B146 79 0xB77C94C2 0x3F9778 0x0B050 80 0xB67EE6EE 0x3F3F8C 0x0AF5C 81 0xB58297D3 0x3EE81C 0x0AE6A 82 0xB40AAEA2 0x7CCBB8 0x2B404 83 0xB21A31A6 0x7B7394 0x2AC90 84 0xB02F0DCB 0x7A1F26 0x2A530 85 0xAE493452 0x78CE62 0x29DE4 86 0xAC6896A4 0x77813E 0x296AC 87 0xAA8D2652 0x7637B2 0x28F8C 88 0xA8B6D516 0x74F1B2 0x2887C 89 0xA6E594CF 0x73AF34 0x28180 90 0xA5195786 0x727030 0x27A94 91 0xA3520F68 0x71349C 0x273C0 92 0xA18FAECA 0x6FFC70 0x26CFC 93 0x9FD22825 0x6EC79E 0x2664C 94 0x9E196E18 0x6D9620 0x25FAC 95 0x9C657368 0x6C67EE 0x25924 96 0x9AB62AFC 0x6B3CFC 0x252A8 97 0x990B87E2 0x6A1544 0x24C40 98 0x97657D49 0x68F0BA 0x245EC 99 0x95C3FE86 0x67CF56 0x23FA4 100 0x9426FF0F 0x66B112 0x23974 101 0x928E727D 0x6595E2 0x23350 102 0x90FA4C8B 0x647DC0 0x22D40 103 0x8F6A8117 0x6368A2 0x22740 104 0x8DDF0420 0x625680 0x2214C 105 0x8C57C9C4 0x614752 0x21B70 106 0x8AD4C645 0x603B10 0x215A0 107 0x8955EE03 0x5F31B2 0x20FE0 108 0x87DB357F 0x5E2B2E 0x20A30 109 0x8664915B 0x5D2780 0x20490 110 0x84F1F656 0x5C269E 0x1FF00 111 0x8383594E 0x5B2880 0x1F97C 112 0x8218AF43 0x5A2D1E 0x1F40C 113 0x80B1ED4F 0x593472 0x1EEA8 114 0x7E9F0606 0xAF894A 0x79AC0 115 0x7BE86FB9 0xABC662 0x77108 116 0x7940BB9E 0xA8181A 0x74838 117 0x76A7980F 0xA47E02 0x72048 118 0x741CB528 0xA0F7AE 0x6F930 119 0x719FC4B9 0x9D84AE 0x6D2F0 120 0x6F307A41 0x9A249C 0x6AD80 121 0x6CCE8AE1 0x96D70C 0x688E0 122 0x6A79AD55 0x939B9C 0x66508 123 0x683199ED 0x9071E6 0x641F0 124 0x65F60A7F 0x8D598A 0x61FA0 125 0x63C6BA64 0x8A5228 0x5FE08 126 0x61A3666D 0x875B64 0x5DD28 127 0x5F8BCCDB 0x8474E2 0x5BD00 128 0x5D7FAD59 0x819E48 0x59D80 129 0x5B7EC8F1 0x7ED742 0x57EB8 130 0x5988E209 0x7C1F76 0x56090 131 0x579DBC56 0x797694 0x54310 132 0x55BD1CDA 0x76DC4A 0x52630 133 0x53E6C9DA 0x745046 0x509F8 134 0x521A8AD7 0x71D23A 0x4EE50 135 0x50582888 0x6F61DC 0x4D340 136 0x4E9F6CD3 0x6CFEDE 0x4B8C8 137 0x4CF022C9 0x6AA8F6 0x49EE8 138 0x4B4A169B 0x685FE0 0x48588 139 0x49AD1597 0x662352 0x46CC0 140 0x4818EE21 0x63F30A 0x45478 141 0x468D6FAD 0x61CEC2 0x43CB8 142 0x450A6ABA 0x5FB63C 0x42578 143 0x438FB0CB 0x5DA934 0x40EB8 144 0x421D1462 0x5BA76C 0x3F878 145 0x40B268FA 0x59B0A6 0x3E2B0 146 0x3EA0ECB7 0xADA6CC 0xF0BA0 147 0x3BF92E67 0xA64A10 0xE6860 148 0x396E41BA 0x9F3D3C 0xDCC00 149 0x36FEEDE6 0x987CEC 0xD3640 150 0x34AA0764 0x9205E0 0xCA6E0 151 0x326E6F61 0x8BD504 0xC1D80 152 0x304B1332 0x85E754 0xB9A00 153 0x2E3EEBD2 0x803A00 0xB1C20 154 0x2C48FD60 0x7ACA48 0xAA380 155 0x2A6856AD 0x759594 0xA3020 156 0x289C10C1 0x709960 0x9C180 157 0x26E34E6E 0x6BD344 0x957A0 158 0x253D3BEA 0x6740FC 0x8F240 159 0x23A90E63 0x62E050 0x89120 160 0x222603A0 0x5EAF28 0x83420 161 0x20B361A6 0x5AAB7C 0x7DB20 162 0x1F50765B 0x56D364 0x785E0 163 0x1DFC9733 0x532508 0x73420 164 0x1CB720DD 0x4F9E9C 0x6E600 165 0x1B7F76F3 0x4C3E74 0x69B20 166 0x1A5503B2 0x4902F0 0x65360 167 0x193737B1 0x45EA80 0x60EC0 168 0x18258999 0x42F3AC 0x5CD00 169 0x171F75E9 0x401D00 0x58E00 170 0x16247EB0 0x3D6524 0x551C0 171 0x15342B57 0x3ACAC8 0x51800 172 0x144E0860 0x384CB0 0x4E0C0 173 0x1371A737 0x35E9A4 0x4ABC0 174 0x129E9DF5 0x33A07C 0x47920 175 0x11D48731 0x317028 0x44880 176 0x111301D0 0x2F5794 0x41A20 177 0x1059B0D3 0x2D55C0 0x3ED80 178 0x0F5257D1 0x54FA30 0xEB980 179 0x0E0CCDEF 0x4DECA8 0xD8080 180 0x0CE248C1 0x477500 0xC6200 181 0x0BD08A3A 0x4186C0 0xB5A80 182 0x0AD583EF 0x3C1688 0xA6980 183 0x09EF5326 0x3719D8 0x98C00 184 0x091C3D37 0x328720 0x8C180 185 0x085AAC36 0x2E5590 0x80780 186 0x07A92BE9 0x2A7D18 0x75D00 187 0x070666F7 0x26F658 0x6C080 188 0x06712461 0x23BA80 0x63100 189 0x05E8451D 0x20C360 0x5AD80 190 0x056AC1F7 0x1E0B40 0x53480 191 0x04F7A993 0x1B8CE8 0x4C600 192 0x048E1E9C 0x194390 0x46080 193 0x042D561B 0x172AC8 0x40380 194 0x03D495F4 0x153E88 0x3AE80 195 0x0383337C 0x137B28 0x36000 196 0x03389230 0x11DD40 0x31880 197 0x02F4228E 0x1061B0 0x2D680 198 0x02B560FC 0x0F05A0 0x29A80 199 0x027BD4C9 0x0DC678 0x26300 200 0x02470F4E 0x0CA1C8 0x23080 201 0x0216AB0E 0x0B9560 0x20200 202 0x01EA4AFA 0x0A9F48 0x1D700 203 0x01C199BE 0x09BD98 0x1B000 204 0x019C4918 0x08EEA0 0x18C00 205 0x017A1147 0x0830D8 0x16B80 206 0x015AB07E 0x0782D0 0x14D00 207 0x013DEA65 0x06E338 0x13180 208 0x012387A7 0x0650E0 0x11800 209 0x010B5587 0x05CAB0 0x10100 210 0x00EAC0C7 0x0A2D70 0x38600 211 0x00C5672A 0x088EF0 0x2F800 212 0x00A5FED7 0x073250 0x27E00 213 0x008B95C2 0x060D30 0x21800 214 0x00756063 0x0516C0 0x1C400 215 0x0062B395 0x044770 0x17C00 216 0x0052FF6B 0x039930 0x14000 217 0x0045CAE1 0x0306A0 0x10C00 218 0x003AB032 0x028B60 0x0E200 219 0x003159CB 0x0223C0 0x0BE00 220 0x00297FB6 0x01CC90 0x0A000 221 0x0022E570 0x018350 0x08600 222 0x001D5819 0x0145B0 0x07000 223 0x0018ACE5 0x0111E0 0x05E00 224 0x0014BFDB 0x00E650 0x05000 225 0x001172B8 0x00C1A0 0x04400 226 0x000EAC0C 0x00A2D0 0x03800 227 0x000C5673 0x0088F0 0x03000 228 0x000A5FED 0x007320 0x02800 229 0x0008B95C 0x0060D0 0x02200 230 0x00075606 0x005170 0x01C00 231 0x00062B39 0x004470 0x01800 232 0x00052FF7 0x003990 0x01400 233 0x00045CAE 0x003070 0x01000 234 0x0003AB03 0x0028B0 0x00E00 235 0x0003159D 0x002240 0x00C00 236 0x000297FB 0x001CD0 0x00A00 237 0x00022E57 0x001830 0x00800 238 0x0001D582 0x001460 0x00800 239 0x00018ACE 0x001120 0x00600 240 0x00014BFE 0x000E60 0x00400 241 0x0001172C 0x000C20 0x00400 242 0x00008000 0x002E80 0x08000 243 0x00002000 0x000B80 0x00000 244 0x00000800 0x000300 0x00000 245 0x00000200 0x000080 0x00000 246 0x00000080 0x000000 0x00000 247 0x00000020 0x000000 0x00000 248 0x00000008 0x000000 0x00000 249 0x00000002 0x000000 0x00000 LOG2F32 Table 1 0x01709C47 0xB7F26D 0x2DCED 2 0x044D8C46 0xB686CB 0x2D1A7 3 0x0724D8EF 0xB520BB 0x2C6A2 4 0x09F6984A 0xB3C01D 0x2BBDE 5 0x0CC2DFE2 0xB264D2 0x2B158 6 0x0F89C4C2 0xB10EBB 0x2A70E 7 0x124B5B7E 0xAFBDBA 0x29CFF 8 0x1507B836 0xAE71B3 0x29328 9 0x17BEEE97 0xAD2A89 0x28989 10 0x1A7111DF 0xABE821 0x2801F 11 0x1D1E34E3 0xAAAA61 0x276E9 12 0x1FC66A0F 0xA9712F 0x26DE6 13 0x2269C369 0xA83C73 0x26513 14 0x25085296 0xA70C13 0x25C71 15 0x27A228DB 0xA5DFFA 0x253FD 16 0x2A375721 0xA4B80F 0x24BB6 17 0x2CC7EDF6 0xA3943C 0x2439A 18 0x2F53FD90 0xA2746D 0x23BAA 19 0x31DB95D0 0xA1588B 0x233E3 20 0x345EC646 0xA04083 0x22C44 21 0x36DD9E2F 0x9F2C40 0x224CD 22 0x39582C79 0x9E1BB0 0x21D7B 23 0x3BCE7FC7 0x9D0EBE 0x2164F 24 0x3E40A672 0x9C055A 0x20F48 25 0x40AEAE89 0x9AFF71 0x20863 26 0x4318A5D5 0x99FCF1 0x201A1 27 0x457E99DB 0x98FDCA 0x1FB00 28 0x47E097DB 0x9801EB 0x1F480 29 0x4A3EACD7 0x970944 0x1EE20 30 0x4C98E58E 0x9613C5 0x1E7DF 31 0x4EEF4E83 0x95215F 0x1E1BC 32 0x5141F3FB 0x943204 0x1DBB6 33 0x5390E204 0x9345A4 0x1D5CE 34 0x55DC246D 0x925C31 0x1D001 35 0x5823C6D1 0x91759E 0x1CA4F 36 0x5A67D492 0x9091DD 0x1C4B9 37 0x5CA858DF 0x8FB0E1 0x1BF3C 38 0x5EE55EB1 0x8ED29D 0x1B9D8 39 0x611EF0CF 0x8DF705 0x1B48E 40 0x635519CF 0x8D1E0B 0x1AF5B 41 0x6587E415 0x8C47A5 0x1AA40 42 0x67B759D6 0x8B73C7 0x1A53D 43 0x69E3851C 0x8AA265 0x1A04F 44 0x6C0C6FC0 0x89D373 0x19B78 45 0x6E322370 0x8906E9 0x196B6 46 0x7054A9B1 0x883CB9 0x1920A 47 0x72740BDB 0x8774DB 0x18D71 48 0x74905320 0x86AF44 0x188ED 49 0x76A98888 0x85EBEB 0x1847D 50 0x78BFB4F4 0x852AC4 0x1801F 51 0x7AD2E11F 0x846BC8 0x17BD5 52 0x7CE3159F 0x83AEED 0x1779C 53 0x7EF05AE4 0x82F429 0x17376 54 0x80FAB93C 0x823B74 0x16F61 55 0x830238D0 0x8184C5 0x16B5D 56 0x8506E1A8 0x80D014 0x1676A 57 0x8708BBAA 0x801D59 0x16387 58 0x8907CE9D 0x7F6C8B 0x15FB4 59 0x8B042225 0x7EBDA2 0x15BF1 60 0x8CFDBDC8 0x7E1096 0x1583E 61 0x8EF4A8ED 0x7D6561 0x15499 62 0x90E8EADE 0x7CBBFB 0x15103 63 0x92DA8AC6 0x7C145B 0x14D7C 64 0x94C98FB4 0x7B6E7C 0x14A03 65 0x96B6009B 0x7ACA56 0x14697 66 0x989FE451 0x7A27E2 0x14339 67 0x9A874193 0x79871A 0x13FE9 68 0x9C6C1F01 0x78E7F7 0x13CA5 69 0x9E4E8325 0x784A73 0x1396E 70 0xA02E746A 0x77AE87 0x13644 71 0xA20BF926 0x77142D 0x13325 72 0xA3E71797 0x767B60 0x13013 73 0xA5BFD5DF 0x75E418 0x12D0C 74 0xA7963A0D 0x754E51 0x12A11 75 0xA96A4A17 0x74BA05 0x12722 76 0xAB3C0BDC 0x74272E 0x1243D 77 0xAD0B8526 0x7395C6 0x12163 78 0xAED8BBA8 0x7305C9 0x11E94 79 0xB0A3B502 0x727731 0x11BCF 80 0xB26C76BC 0x71E9F8 0x11914 81 0xB433064B 0x715E1A 0x11664 82 0xB5F76913 0x70D393 0x113BD 83 0xB7B9A45E 0x704A5C 0x11120 84 0xB979BD69 0x6FC271 0x10E8C 85 0xBB37B959 0x6F3BCE 0x10C02 86 0xBCF39D45 0x6EB66D 0x10981 87 0xBEAD6E2D 0x6E324B 0x10709 88 0xC0653103 0x6DAF63 0x10499 89 0xC21AEAA6 0x6D2DB1 0x10232 90 0xC3CE9FE4 0x6CAD30 0x0FFD4 91 0xC5805579 0x6C2DDC 0x0FD7E 92 0xC7301011 0x6BAFB1 0x0FB30 93 0xC8DDD449 0x6B32AA 0x0F8EA 94 0xCA89A6AC 0x6AB6C5 0x0F6AC 95 0xCC338BB7 0x6A3BFD 0x0F475 96 0xCDDB87D6 0x69C24F 0x0F247 97 0xCF819F66 0x6949B5 0x0F01F 98 0xD125D6B7 0x68D22E 0x0EDFF 99 0xD2C83209 0x685BB5 0x0EBE7 100 0xD468B58C 0x67E646 0x0E9D5 101 0xD6076565 0x6771DF 0x0E7CA 102 0xD7A445A9 0x66FE7B 0x0E5C6 103 0xD93F5A60 0x668C18 0x0E3C9 104 0xDAD8A784 0x661AB1 0x0E1D2 105 0xDC703104 0x65AA44 0x0DFE2 106 0xDE05FAC0 0x653ACE 0x0DDF8 107 0xDF9A088A 0x64CC4B 0x0DC14 108 0xE12C5E2B 0x645EB9 0x0DA37 109 0xE2BCFF5E 0x63F213 0x0D85F 110 0xE44BEFD0 0x638658 0x0D68E 111 0xE5D93326 0x631B84 0x0D4C2 112 0xE764CCF7 0x62B195 0x0D2FC 113 0xE8EEC0CE 0x624887 0x0D13C 114 0xEA77122B 0x61E058 0x0CF82 115 0xEBFDC485 0x617904 0x0CDCC 116 0xED82DB45 0x61128A 0x0CC1D 117 0xEF0659CC 0x60ACE7 0x0CA72 118 0xF088436D 0x604818 0x0C8CD 119 0xF2089B75 0x5FE41A 0x0C72C 120 0xF3876524 0x5F80EA 0x0C591 121 0xF504A3AF 0x5F1E87 0x0C3FB 122 0xF6805A44 0x5EBCEE 0x0C26A 123 0xF7FA8C05 0x5E5C1D 0x0C0DD 124 0xF9733C0C 0x5DFC11 0x0BF55 125 0xFAEA6D67 0x5D9CC7 0x0BDD2 126 0xFC60231E 0x5D3E3E 0x0BC53 127 0xFDD4602E 0x5CE073 0x0BAD9 128 0xFF47278B 0x5C8364 0x0B964

Different tables can be employed for different example transcendental functions. One set of tables for a particular transcendental function can have a fixed distance between entry points of an input, and another set of tables for another transcendental function can have a variable distance between entry points of the input.

FIG. 6 is a block diagram of an example data path of an example logarithmic execution unit for determining logarithmic transcendental functions. For example, the logarithmic accelerator 600 is arranged to compute a result of a logarithmic function (e.g., log₂(x)) in response to an input operand. The input operand is a floating-point number floating-point number in the form of a “1.M” form (where “M” is the mantissa) and having a value between 1 and 2.

The logarithmic accelerator 600 described herein is arranged to compute a floating point logarithm in response to curve-fitting including quadratic interpolation to generate the logarithmic result. The mantissa (e.g., for addressing a segment in the range [1.0, 2.0]) is represented by a number of equally spaced linear approximating segments. The coefficients for the curve fit can be derived using a least-mean-square approach.

In operation, the logarithmic accelerator 600 reads an input operand 610, where the input operand 610 includes a sign bit 611, exponent bits 612 and a mantissa 613. The mantissa 613 is segmented into most-significant bits (MSBs) 614 and least-significant bits (LSBs) 615. In the example, the input operand 610 includes 32 bits (e.g., as a range of [31:0]). The input operand 610 is parsed to detect any formatting errors by exception detection logic 616 and detected errors are reported as indicated by an error_result module 617.

Module 651 is arranged to generate a 15-bit number in response to the LSBs (e.g., 16 bits) 615 of the mantissa 613 of the input operand 610. For example, the module 651 is arranged to determine the absolute value of the result of subtracting the hexadecimal number 0x8000 from the LSBs 615. The output of the module 651 is used to, for example, interpolate values determined from the first table 634 and the second table 655 as described hereinbelow.

A first circuit 630 is arranged to generate a linear term of the logarithmic function in response to the LSBs 615 of the mantissa 613 of the input operand 610 and in response to a first table value S1 that is retrieved from a first table 634 in response to a first index Index1 generated in response to the MSBs 614 of the mantissa 613 of the input operand 610. As indicated by a module 632, the 15-bit output of the module 651 is left-shifted 10 times to produce a 25-bit number dx. The lowest seven bits of the bits [30:16] of the input operand can be selected for address the 128 entries of the LOG 2F32 table of Table 2, for example. (In various examples, the exponent only can be used to generate an index, although extra circuitry would otherwise be required to accommodate non-linearity in the logarithmic transfer function.) The notation dx<<10 represents a shift operation of 10 bits. As indicated by a module 633, the number dx is multiplied by the first table value S1 and the result is a 49-bit number S1*dx. The 49-bit number is truncated to a 26-bit number as indicated by module 635, with the least significant 22 bits being discarded. The 26 bit number or term S1*dx is presented to a combiner 660.

A second circuit 650 is arranged to generate a quadratic term of the logarithmic function in response to the LSBs 615 of the mantissa 613 of the input operand 610 and in response to a second table value S2 that is retrieved from a second table 655 in response to a second index Index2 generated in response to the MSBs 614 of the mantissa 613 of the input operand 610. The lowest seven bits of the bits [30:16] of the input operand can be selected for address the 128 entries of the LOG 2F32 table of Table 2, for example. As described hereinabove, the module 651 is arranged to generate a 15-bit number dx in response to the absolute value of the 16-bit LSB number 615 minus the hexadecimal number 0x8000. The number dx is squared to produce a 29-bit number dx*dx as indicated by a module 652.

The 29-bit number is truncated to an 18-bit number as indicated by module 653, with the least significant 11 bits being discarded (e.g., by truncation). As indicated by a module 654, the truncated 18-bit number dx*dx is multiplied by the second table value S2 and the result is a 38-bit number S2*dx*dx. The 38-bit number is truncated to a 20-bit number as indicated by module 656, with the least significant 18-bits being discarded. The 20-bit term S2*dx*dx is presented to the combiner 660.

A third circuit 680 is arranged to generate (e.g., to output) a constant term of the logarithmic function in response to the LSBs 615 of the mantissa 613 of the input operand 610 and in response to a third table value Y0 that is retrieved from a third table 684 in response to a third index Index3 generated in response to the MSBs 614 of the mantissa 613 of the input operand 610. The lowest seven bits of the bits [30:16] of the input operand can be selected for address the 128 entries of the LOG 2F32 table of Table 2, for example. As indicated by a module 681, the terms S1*dx, S2*dx*dx obtained from the combiner 660 and a rounding up constant 1 are added to produce a first result R1. The 26-bit result R1 is truncated to a 25-bit result R1 as indicated by modules 670, 682, with the least significant bit being discarded. As indicated by a module 683, the third table value Y0 is added to the 25 bit result R1 to produce a second result R2. The exponent bits 612 of the input operand 610 are added to the result R2 minus a number 127 as indicated by a module 685. The result R2 is the value of the logarithmic function of the input operand 610. Accordingly, the third circuit 680 is arranged to generate a mantissa of an output operand (the result R2) in response to a sum of the linear term, the quadratic term, and the constant term (in possible results, any of the linear term, the quadratic term, and the constant terms can have a value of zero). Additionally, exception detection logic and output exponent and mantissa adjustment is performed as indicated by a module 686 on the result R2 to identify data failures.

FIG. 7 is a block diagram of an example data path of an example exponentiation execution unit for determining exponentiation transcendental functions. For example, the exponentiation accelerator 700 is arranged to compute a result of an exponential function (e.g., 2^(−|x|)) in response to an input operand. The exponentiation accelerator 700 described herein is arranged to compute a floating point exponential in response to curve-fitting including quadratic interpolation to generate the exponential result. The floating point number is represented by a number of non-equally spaced linear approximating segments. The non-equally spaced segments (e.g., slices) are addressed by the mantissa and exponent of the input operand for approximating a value of the transcendental function. The tables for evaluation of the exponential function are different from the tables employed for evaluation of the logarithmic function.

In operation, the exponentiation accelerator 700 reads an input operand 710 into the logarithmic accelerator 700, where the input operand 710 includes a sign bit, 711, exponent bits 712 and a mantissa 713. The mantissa 713 is segmented into most-significant bits (MSBs) 714 and least-significant bits (LSBs) 715 (e.g., where X can be any value from 22 to 1). In the example, the input operand 710 includes 32 bits (e.g., as a range of [31:0]). The input operand 710 is evaluated to detect any exception conditions (e.g., infinity, NaN (not a number), and Denormal) by exception detection logic 716 and detected errors are reported as indicated by an error_result module 717.

Module 751 is arranged to generate a 15-bit number in response to the LSBs (e.g., 16 bits) 715 of the mantissa 713 of the input operand 710. For example, the module 751 is arranged to determine the absolute value of the result of subtracting the hexadecimal number 0x8000 from the LSBs 715. The output of the module 751 is used to, for example, interpolate values determined from the first table 734 and the second table 755 as described hereinbelow.

A first circuit 730 is arranged to generate a linear term of the exponential function in response to the LSBs 715 of the mantissa 713 of the input operand 710 and in response to a first table value S1 that is retrieved from a first table 734 in response to a first index Index1 generated in response to the MSBs 714 of the mantissa 713 as well as the exponent (bits 30:23 of the exponent) 712. The lowest eight bits of the bits [30:16] of the input operand can be selected for address the 249 entries of the IEXP2F32 table of Table 2, for example. As indicated by a module 732, the LSBs 715 of the mantissa 713 are left-shifted 10 bits to produce a 25-bit number dx. (The notation dx<<10 represents a shift operation by 10 bits.) As indicated by a module 733, the number dx is multiplied by the first table value S1 and the result is a 49-bit number S1*dx. The 49-bit number is truncated to a 26-bit number as indicated by module 735, with the least significant 22 bits being discarded. The 26-bit term S1*dx is presented to a combiner 760.

A second circuit 750 is arranged to generate a quadratic term of the exponential function in response to the LSBs 715 of the mantissa 713 of the input operand 710 and in response to a second table value S2 that is retrieved from a second table 755 in response to a second index Index2 generated in response to the MSBs 714 of the mantissa 713 as well as the exponent 712. The lowest eight bits of the bits [30:16] of the input operand can be selected for address the 249 entries of the IEXP2F32 table of Table 2, for example. As described hereinabove, the module 751 is arranged to generate a 15-bit number dx in response to the absolute value of the 16-bit LSB number minus the hexadecimal number 0x8000. The number dx is squared to produce a 29-bit number dx*dx as indicated by a module 752.

The 29-bit number is truncated to an 18-bit number as indicated by module 753, with the least significant 11 bits being discarded. As indicated by a module 754, the truncated 18-bit number dx*dx is multiplied by the second table value S2 and the result is a 38-bit number S2*dx*dx. The 38-bit number is truncated to a 20-bit number as indicated by module 756, with the least significant 18 bits being discarded. The 20-bit term S2*dx*dx is presented to the combiner 760.

A third circuit 780 is arranged to generate a constant term of the exponential function in response to the LSBs 715 of the mantissa 713 of the input operand 710 and in response to a third table value Y0 that is retrieved from a third table 784 in response to a third index Index3 generated in response to the MSBs 714 of the mantissa 713 as well as the exponent 712. The lowest eight bits of the bits [30:16] of the input operand can be selected for address the 249 entries of the IEXP2F32 table of Table 2, for example. As indicated by a module 781, the terms S1*dx, S2*dx*dx obtained from the combiner 760 and a rounding up constant 1 are added to produce a first result R1. The 26-bit result R1 is truncated to a 25-bit result R1 as indicated by modules 770, 782, with the least significant bit being discarded. As indicated by a module 783, the third table value Y0 is added to the 25-bit result R1 to produce a result R2. The result R2 is the value of the exponential function of the input operand 710. Accordingly, the third circuit 780 is arranged to generate a mantissa of an output operand (the result R2) in response to a sum of the linear term, the quadratic term, and the constant term. Additionally, exception detection logic and output exponent and mantissa adjustment is performed as indicated by a module 786 on the result R2 to identify data failures.

With continuing reference to the preceding figures, a process and related method of operating an apparatus to compute a value of a transcendental function have been introduced herein. In one embodiment, the method includes generating a linear term of a transcendental function in response to least significant bits of a mantissa of an input operand and in response to a first table value that is retrieved from a first table in response to a first index generated in response to most significant bits of the mantissa of the input operand. The method also includes generating a quadratic term for the transcendental function in response to least significant bits of the mantissa of the input operand and in response to a second table value that is retrieved from a second table in response to a second index generated in response to most significant bits of the mantissa of the input operand. The method further includes generating a constant term for the transcendental function in response to least significant bits of the mantissa of the input operand and in response to a third table value that is retrieved from a third table in response to a third index generated in response to most significant bits of the mantissa of the input operand. The method also includes generating an output operand in response to a sum of the linear term, the quadratic term, and the constant term and/or approximations thereof.

In an embodiment, the transcendental function is an exponential function, wherein the first index is further generated in response to exponent of the input operand and MSBs of mantissa, wherein the second index is further generated in response to the exponent and MSB of mantissa of the input operand, and wherein the third index is further generated in response to the exponent and MSBs of mantissa of the input operand.

In an embodiment, the output operand for the transcendental function is generated in response to quadratic approximations of the linear term, the quadratic term, and the constant term, and wherein the quadratic approximations of the linear term, the quadratic term, and the constant terms are respectively generated in response to the first circuit, the second circuit, and the third circuit are arranged to truncate and discard lower order bits to produce, respectively the linear term, the quadratic term, and the constant term

Modifications are possible in the described examples, and other examples are possible, within the scope of the claims. 

What is claimed is:
 1. An apparatus, comprising: an input configured to receive an input operand having a mantissa, wherein the mantissa includes a set of most significant bits (MSBs) and a set of least significant bits (LSBs); a first circuit configured to: retrieve a first table value from a first table based on a first index based on the set of MSBs; and generate a linear term based on the set of LSBs and the first table value; a second circuit configured to: retrieve a second table value from a second table based on a second index based on the set of MSBs; and generate a quadratic term based on the set of LSBs and the second table value; and a third circuit configured to: retrieve a third table value from a third table based on a third index based on the set of MSBs; generate a constant term based on the set of LSBs and the third table value; and provide an output operand based on the linear term, the quadratic term, and the constant term.
 2. The apparatus of claim 1, wherein: the third circuit is further configured to generate a sum of the linear term, the quadratic term, and the constant term; and the output operand is based on the sum.
 3. The apparatus of claim 1, wherein: the first table, the second table and the third table comprise a unified table.
 4. The apparatus of claim 1, wherein: the input operand includes an exponent.
 5. The apparatus of claim 4, wherein: the first index is further based on the exponent; the second index is further based on the exponent; and the third index is further based on the exponent.
 6. The apparatus of claim 1, wherein: the output operand includes a mantissa.
 7. The apparatus of claim 1, wherein: the first circuit and the second circuit are configured to operate in parallel during a first time interval; and the third circuit is configured to operate during a second time interval.
 8. The apparatus of claim 1, wherein: the third circuit is configured to generate the output operand as an exponential function.
 9. The apparatus of claim 8, wherein: the first, second, and third tables include sequences of non-equally spaced segments of values.
 10. The apparatus of claim 9, wherein: the first, second, and third table values are associated with endpoints of each of the non-equally spaced segments of the first, second, and third tables.
 11. The apparatus of claim 9, wherein: spacing between segments is determined based on a least mean-square analysis of a regression of interval spacing and table lengths.
 12. A system, comprising: a memory configured to receive and store an input operand having a mantissa that includes a set of most significant bits (MSBs) and a set of least significant bits (LSBs); and an execution unit configured to generate as an approximation of a transcendental function based on the input operand stored and retrieved from the memory, comprising: a first circuit configured to: retrieve a first table value from a first table based on a first index based on the set of MSBs; and generate a linear term based on the set of LSBs and the first table value; a second circuit configured to: retrieve a second table value from a second table based on a second index based on the set of MSBs; and generate a quadratic term based on the set of LSBs and the second table value; and a third circuit configured to: retrieve a third table value from a third table based on a third index based on the set of MSBs; generate a constant term based on the set of LSBs and the third table value; generate a sum of the linear term, the quadratic term, and the constant term; and provide an output operand based on the sum.
 13. The system of claim 12, wherein: the input operand includes an exponent.
 14. The system of claim 13, wherein: the first index is further based on the exponent; the second index is further based on the exponent; and the third index is further based on the exponent.
 15. The system of claim 12, wherein: the third circuit is configured to generate the output operand as an exponential function; and the first, second, and third tables include sequences of non-equally spaced segments of values.
 16. A method, comprising: receiving, by an execution unit, an input operand that includes a mantissa, wherein the mantissa includes a set of most significant bits (MSBs) and a set of least significant bits (LSBs); retrieving, by the execution unit, a first table value from a first table based on a first index based on the set of MSBs; generating, by the execution unit, a linear term based on the set of LSBs and the first table value; retrieving, by the execution unit, a second table value from a second table based on a second index based on the set of MSBs; generating, by the execution unit, a quadratic term based on the set of LSBs and the second table value; retrieving, by the execution unit, a third table value from a third table based on a third index based on the set of MSBs; generating, by the execution unit, a constant term based on the set of LSBs and the third table value; and outputting, by the execution unit, an output operand based on the linear, quadratic, and constant terms.
 17. The method of claim 16, wherein: the first table, the second table, and the third tables include slices having constant spacing between slices.
 18. The method of claim 16, wherein: the first table, the second table, and the third tables include slices having variable spacing between slices.
 19. The method of claim 16, wherein: the first table, the second table, and the third table comprise a unified table.
 20. The method of claim 16, wherein: the output operand is further based on a summation, by the execution unit, of the linear term, the quadratic term, and the constant term. 